Problem: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{-9t^2 - 36t + 45}{-4t^2 + 20t + 200}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {-9(t^2 + 4t - 5)} {-4(t^2 - 5t - 50)} $ $ n = \dfrac{9}{4} \cdot \dfrac{t^2 + 4t - 5}{t^2 - 5t - 50} $ Next factor the numerator and denominator. $ n = \dfrac{9}{4} \cdot \dfrac{(t + 5)(t - 1)}{(t + 5)(t - 10)}$ Assuming $t \neq -5$ , we can cancel the $t + 5$ $ n = \dfrac{9}{4} \cdot \dfrac{t - 1}{t - 10}$ Therefore: $ n = \dfrac{ 9(t - 1)}{ 4(t - 10)}$, $t \neq -5$